We now introduce the preconditioners we use to
solve (REF ). We express the solvers in terms of
space decompositions [1]}, which we summarize briefly here. Given
a discrete space \(V^k_{h,p}\) , the preconditioner is induced by a particular choice of
how to write it as a sum of (smaller) function spaces:
\( V^k_{h,p} = \sum _i V_i.\) 
